Mean particle travel time

For the hydraulic base case, constant hydraulic apertures and fracture densities were considered. Homogeneous hydraulic conductivity tensors and porosities were applied to each formation. The case with a constant hydraulic aperture of 10 im and medium fracture density for all formations is illustrated in Figure 5, where the streamlines and the particle travel times are shown. The time of travel between each marker is 1000 days. Owing to the low effective porosity values and the relatively high fracture density, particle travel times through the host rock are fast. The mean particle travel time from the repository to the seabed is only 123 years, when a constant hydraulic aperture of 10 pm is used. 

The results for the particle travel times for various constant hydraulic apertures and fracture densities from the repository to the seabed are presented graphically in Figure 4. For the low fracture density case, the hydraulic conductivity estimated for a block size of 25 m x 25 m has been used even though this size does not correspond to the REV, which is estimated to be greater than 100 m x 100 m. This block size does allow, however, a calculation of the mechanical closure of the fracture apertures for the HM case and therefore a comparison of the results between the two cases for low-density conditions. The results of the continuum model based on constant hydraulic apertures display very rapid mean particle travel times from the repository to the seabed. For example, for the low and high fracture density networks adopting a constant hydraulic aperture of 10 pm, particle travel times from the repository to the seabed are 580 years and 106 years respectively. A doubling of the aperture increases the conductivity by a factor of... 

Figure 6. Hypothetical host rock with streamlines and particle travel times for the mean HM base case.

The length scale for describing the spatial variation of the pollutant species, L, is large when compared with the turbulent mean free path, /, a distance traveled by a particle in time Ti. Assuming the root-mean-square (rms) turbulent velocity of the particle is , this condition can be expressed as L > / = mJl. 

The basic Gaussian plume dispersion parameters are ay and a. The essential theoretical result concerning the dependence of these parameters on travel time is for stationary, homogeneous turbulence (Taylor, 1921). Consider marked particles that are released from the origin in a stationary, homogeneous turbulent flow with a mean flow in the x direction. The y component, y, of the position of a fluid particle satisfies the equation... 

It has been pointed out over the years that the simple exponential function of the form where / is travel time from the source, appears to approximate the Lagrangian velocity autocorrelation function R t) rather well (Neumann, 1978 Tennekes, 1979). If R(t) = exp(-l/r), then the mean square particle displacement is given by (Taylor, 1921)... 

The question arises How far will a particle travel in a given time interval The average distance a particle travels is given by mean square displacement evaluated as follows The position of a particle along the z axis after i steps z is... 

The mean square distance that a particle travels depends upon the time of travel in the following manner. The number of steps N obviously increases with time and is proportional to it, i.e.. 

The movement of solute particles in soils does not generally follow Brownian motion because soil at any scale consists of correlated units. A particle traveling faster than the mean at some instant is likely to still be traveling faster than the mean some later time (Benson, 1998). One may say that particles have a memory , and the memory of particles is the critical feature required for the occurrence of non-Fickian diffusion (Kin/clhach. 1987). Further, the underlying distribution... 

The mean position of the distribution at time t is ut, just the distance a tracer molecule has traveled over a time t at the mean fluid velocity u. The variance of the mean concentration distribution, cr (r), is just the variance of X(/,x). This result makes sense since X(r,x) is just the random distance that a fluid particle travels between times x and t, and this distance is precisely that which a tracer molecule travels. 

On the way toward the wall having a surface area A, the particle transports momentum mvx, and after colliding with the waU, it transports momentum —mvx in the opposite direction (Fig. 10.5). The wall has absorbed the difference 2mox during the collision. Between two collisions with waU A, the particle travels the distance 2a. In the time span At, it covers the distance Vx At. This means that in the time span At, the number of collisions is given by... 

The assertion that gas molecules occupy no volume may seem at odds with reality because we know that all matter occupies space. What is actually implied by this postulate The essential idea is that, compared to the volume of empty space between particles, the volume of the particles themselves is not significant. One way to assess the validity of this assertion is to define the mean free path of the particles. The mean free path is the average distance a particle travels between collisions with other particles. In air at room temperature and atmospheric pressure, the mean free path is about 70 nm, a value 200 times larger than the typical radius of a small molecule like N2 or O2. (For comparison, the molecules in a liquid typically have a mean free path roughly the same as a molecular radius.) When we consider the cubic relationship of volume to distance, the difference in volume is on the order of 200 or 8 x 10. So the volume of empty space in a gas at room tem-peramre and pressure is on the order of 1 million times greater than the volume of the individual molecules the assumption in the kinetic theory that the volume of the molecules is negligible seems reasonable under these conditions. 

Finally, consider a box having dimensions a X b X c in the x, y, and z dimensions, as shown in Figure 19.2. What is the time amount in equation 19.6 When the particle is not colliding with the wall, no force is being exerted on it, nor by it on anything else (that s one of the postulates). The time can be as long as it takes for the particle to start at one wall, travel in the x dimension to collide with the other wall, then travel all the way back to the opposite wall. This means that the particle travels twice the x dimension, or a distance of 2a. From the definition of velocity we have... 

In its original form [18,19], the SRD algorithm was not Galilean invariant. This is most pronounced at low temperatures or small time steps, where the mean free path, A = At k T/m, is smaller than the cell size a. If the particles travel a distance between collisions which is small compared to the cell size, essentially the same... 

The average time between collisions is then v and in this time tlie particle will typically travel a distance X, the mean free path, where... 

This discussion of geometric effects ignored the attenuation of radiation by material through which the radiation must travel to reach the receptor. The number of particles, dN, penetrating material, equals the number of particles incident N times a small penetration distance, dx, divided by the mean free path length of the type of particle in the type of material (equation 8.3-8). Integrating gives the transmission coefficient for the radiation (equation 8.3-9). 

Compression occurs when the space is decreased between the molecules. Less volume means that each particle has a shorter distance to travel, thus proportionately more collisions occur in a given span of time, resulting in a higher pressure. Air compressors are designed to generate particular pressures to meet specific application requirements. 

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